The two-change problem and calculus students’ thinking about direction and path

• I describe students’ understanding of rate of change in two and three dimensions.
• I present an overview of clinical interviews and a subsequent sequence of exploratory teaching interviews.
• I explain the two-change problem as a problem students encounter in thinking about rate in three dimensions.
• I illustrate how the two-change problem can necessitate directional derivative and path.

The purpose of this paper is to propose the two-change problem as an important conceptual issue that students experience as they reason about the rate of change of multivariable functions. This paper presents the results of interviews to illustrate how students conceived of the two-change problem and attempted to resolve it. In doing so, they developed initial notions of the dependence of rate of change on direction and path. The paper closes by discussing the implications of the two-change problem for engendering useful ways of thinking about rate of change in three or more dimensions.